It is a theorem of Bilu that algebraic numbers of large degree and small Weil height tend to "cluster" around the unit circle in the complex plane in the sense that the probability measure with point masses at each of the Galois conjugates converges weakly to the normalized Haar measure of the unit circle. Quantitative results in this direction have drawn upon classical theorems regarding the angular equidistribution of roots of polynomials. However, little is known about the limiting radial distribution of such Galois orbits. We make and present evidence for a conjecture on the radial distribution and illustrate what light it might shed on this problem. This is a candidacy exam.

We will discuss some of the basics of conformal geometry in a familiar setting, namely the uniformization of surfaces. The uniformization theorem is proved by solving the analogue of the Yamabe problem in two dimensions, which is the same as solving a semilinear elliptic pde; the properties of this pde (specifically, compactness) depend heavily on the topology of the manifold, and proving the theorem is much harder when the Euler characteristic is positive.

I'll discuss an approach to perturbative renormalisation of certain quantum field theories, based on the Wilsonian renormalisation group flow. This approach applies to gauge theories in the BV formalism, but in this talk I'll concentrate on scalar field theories.

We study smooth families of diffeomorphisms of three-space that have invariant manifolds arising from a perturbation of a saddle connection. In this context, chaotic transport appears naturally. This version of phase space transport is based on structures that are formed of pieces of normally hyperbolic invariant manifolds, called lobes. In this talk, we will explore the topology of lobes and possible measures of their complexity.

I'll try to explain the (surprisingly strong) connections between the topology of the moduli spaces of surfaces and the homological algebra of non-commutative Frobenius algebras.

Let denote the multiplicative group of nonzero algebraic numbers modulo its subgroup of roots of unity. The absolute logarithmic Weil height is well defined on and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of $L^1(Y, B, \lambda)$, where is a certain totally disconnected, locally compact space, is the -algebra of Borel subsets of , and is a certain measure satisfying an invariance property with respect to the absolute Galois group. This is joint work with Daniel Allcock.

Abstract : I'll explain how perturbative Chern-Simons theory gives an algebraic structure on the cohomology of manifolds, enriching the commutative-infinity structure encoding the rational homotopy type.

We characterize models where the mutual fund theorem holds true in terms of the replicability of options written on the numeraire portfolio (the wealth process of the logarithmic utility maximizer). The characterization works in general semimartingale models, not only in the Brownian Motion case. The presentation is based on joint work with W. Schachermayer and Erik Taflin

The largest simple finite sporadic group, the Monster M, with order $10^{54}$ is constructed via Kac-Moody algebras and Vertex Operators. It might have relation also (Witten) with the 3-dim black hole BTZ. In the talk I shall try to describe the group in relation to physics constructions.

It was known to Lagrange that every positive integer is the sum of four squares. After attending this talk (or after reading its title), you will know this fact, as well. We will discuss the proof of Lagrange's Theorem and talk about the more general Waring Problem, which asks if every integer n is the sum of some fixed number s of kth powers. The answer to this problem may surprise you. But it likely will not.

Using Weil’s height (the logarithmic and absolute height), the famous Lehmer problem reads as follows: does there exist a constant such that for any algebraic number , $a\neq 0$ and not a root of unity, we have $h(a) \geq c/[Q(a):Q]$? I will present a proof given by Francesco Amoroso and Roberto Dvornicich that algebraic numbers in abelian extensions have (logarithmic and absolute) Weil height greater than $(\log 5)/12$. The proof rests on elementary congruences for elements lying in cyclotomic fields, then the result follows for abelian extensions via Kronecker-Weber. As a corollary, we prove a slightly weaker version of Smyth's theorem - which gave a lower bound for algebraic numbers whose minimal polynomials were nonreciprocal. Smyth's proof of this used power series and tools from complex analysis. You might want to look at the strong approximation theorem and Kronecker-Weber before the talk. If you have seen a height talk before, then the rest should be fairly accessible.

We will finish the proof of the Uniformization theorem. Last week we proved that any metric on a compact orientable 2 surface of non-positive Euler characteristic is conformal to a metric of constant (non-positive) Gauss curvature. To prove the theorem when the underlying manifold is topologically the 2-sphere we will reduce the proof to a problem in harmonic maps which is then solved using an important result of Karen Uhlenbeck, this result deals with the removal of singularities of harmonic maps with finite energy.

I am going to talk about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. We consider the construction via three gluings, that is, gluing together two 3-manifolds with incompressible boundary, gluing together the boundary components of surface $\times$ I, and gluing a handlebody to the boundary of a 3-manifold. Then we investigate the Heegaard genus of the resulting manifolds, which is usually less than that of the manifolds before gluing.

Gaussian beams are approximate solutions to hyperbolic partial differential equations that are concentrated on a curve in space-time. We will use superpositions of such solutions to compute the stationary in time wave field that results from steady air flow over a mountain. We will derive the system of equations that governs these mountain waves as a linearization of the basic equations of fluid dynamics and show that this system is well-posed.

A complex K3 surface is called singular if it has the maximal Picard number 20. Singular K3 surfaces behave in many ways like elliptic curves with complex multiplication. After reviewing the classical theory we will sketch new results in this direction. Particular concern will be paid to fields of definition and modular forms.

The Perron-Frobenius Theorem is often proved using Brouwer's Fixed-Point Theorem. The proof of the latter relies on ideas from analysis and topology. We prove one of the main results of the Perron-Frobenius Theorem - that a positive matrix has a real and positive eigenvalue - using Sperner's Lemma, which is a result proved by employing simple counting techniques. We construct a generalized Sperner labeling of a space transformed linearly by a positive matrix and show that there is some vector whose image has the same direction under the transformation.

We introduce the "Seifert Surgery Network" consisting of all the integral Dehn surgeries on knots in the 3-sphere yielding Seifert fiber spaces, where Seifert fiber spaces may have fibers of indices zero as a degenerate case. We will start with some general results and then discuss the connectivity of the network. In particular, we will discuss which surgeries on torus knots can be "spreaders" in the Seifert Surgery Network. (joint work with Arnaud Deruelle and Katura Miyazaki)

I'll show how the Howe-Moore theorem can be used to prove a pointwise ergodic theorem for PSL_2(R). Suprisingly, this generalizes to arbitrary word hyperbolic groups. I'll also talk about how to obtain a pointwise ergodic theorem for actions of word hyperbolic groups on finite sets.

Given a field k and a collection of polynomials in k[x] that span a subspace S of k[x], the Wronskian of those polynomials is defined to be the determinant of a matrix formed from the polynomials and their derivatives. Given a different collection of polynomials that also span S, the Wronskian will be the same up to a multiplicative constant, which suggests that we should consider the Wronskian as a map from subspaces to a projective space. In this talk I will introduce the Grassmann coordinates that index subspaces, and outline a construction of the Wronskian from the Grassmann coordinates that requires no differentiation.

I will discuss the Atiyah-Singer index theorem and sketch a proof. Time permitting, I will describe some of the constructions arising from it.

This is a joint work with Guy Barles (Tours) and Emmanuel Chasseigne (Tours). We consider second order elliptic equations that can be degenerate, non-linear and can contain singular integral terms. The most important example is the Hamilton-Jacobi-Bellman equation satisfied by the value function associated with a stochastic control of jump processes. We investigate the Dirichlet problem and the H\"{o}lder regularity of the solution. In particular, we exhibit structure condition on the singular measure that ensures that the solution coincides with the Dirichlet datum at the boundary. We also explain how to get existence and uniqueness of such a solution.

I will discuss what it means for two hyperbolic manifolds to be commensurable and then talk about the commensurability classes of hyperbolic two-component two-bridge links.

We discuss the problem of constructing elements of high order in finite fields of large degree over their prime field. We prove that for points on a plane curve, one of the coordinates has to have high order. We also discuss a conjecture of Poonen for subvarieties of semiabelian varieties for which our result is a weak special case.

I will prove that each eigenvalue of the Dirichlet Laplacian is strictly less than the corresponding eigenvalue of the Stokes operator on a bounded domain in the plane.

Let be a positive integer. Define $T(n)= \left\{ \begin{array}{cc} \frac{n}{2}, &\mbox{ if n is even} \\ \frac{3n+1}{2}, &\mbox{ if n is odd} \end{array}\right\}$. A well known conjecture asserts that the sequence obtained by iterating always reaches the integer 1 and this happens independent of which positive integer begins the sequence. We will discuss some basic results and methods for attacking this conjecture.

(This is joint with Nonlinear Dynamics.)

I ll talk about some work by Moser on the stability of foliations on a torus. This boils down to showing the existence of solutions to a degenerate elliptic PDE, which is done by adding a small term to the equation and showing existence independent of the size of the extra term.

Given a collection of linearly independent polynomials, we form the Wronskian by taking the determinant of a matrix involving those polynomials and their derivatives. In this talk, I will discuss some properties of the Wronskian, and prove an identity that gives the Wronskian in terms of certain Grassmann coordinates that we may associate with the polynomials in question. This is a candidacy talk.

This is a continuation of last's week talk

In this talk we will present a joint work with Pierre Germain and Gigliola Staffilani on regularity of solutions to the Navier-Stokes equations evolving from small initial data in a critical space in . More precisely, in 2001 H. Koch and D. Tataru proved the existence of global in time solutions to the incompressible Navier-Stokes equations in corresponding to the initial data small enough in $BMO^{-1}$. We prove that under certain smallness condition on the initial data in $BMO^{-1}$, the solutions constructed by Koch and Tataru are more regular. As a consequence, we obtain a decay estimate in time for any space derivative, and space analyticity of the solution. Also as an application of our regularity theorem, we prove a regularity result for self-similar solutions.

This talk explores Thurston's Hyperbolic Dehn Surgery Theorem as exemplified by Hoste and Shanahan's analysis of twist knots being the result of (1/n)-surgery on one component of the Whitehead link.

Deciding the non-emptiness of an algebraic set over a finite field (given as a system of polynomial equations) is one of the most basic NP-complete problems. What is less clear is whether the one variable case can be done in polynomial time. Also, the algorithmic complexity of the analogous question over the complex numbers (for arbitrary dimension) remains unknown. After reviewing some background and concrete examples, we give partial answers to both questions: (1) If one can decide whether sparse polynomials in one variable have roots over finite fields in randomized polynomial-time, then a currently disbelieved complexity collapes occurs (NP is in BPP). In other words, restricting to one variable is unlikely to be doable even in randomized polynomial-time. This answers a question posed by David A. Cox around 2004. (2) One can decide (for most inputs) whether polynomial systems vanish at torsion points in polynomial time iff P=NP. Put another way, this means that torsion points problems have sufficiently low complexity to be closely related to the famous P=?NP question. Independent of any complexity-theoretic aspects, the underlying techniques provide a practical number-theoretic alternative to resultant and Grobner-basis techniques. We also highlight a connection to a (former) conjecture of Lang on torsion points. We assume no background in complexity theory.

This is a mini-course consisting of five lectures, all of them in RLM 11.176 Lecture 1: Fri, Sept 28, 10:30-12:00 Lecture 2: Mon, Oct 1, 2-3:30 Lecture 3: Fri, Oct 5, 10:30-12 Lecture 4: Mon, Oct 22, 2-3:30 Lecture 5, Fri, Oct 26, 10:30-12:00

I will give the condition for an integer to be a sum of two squares and present two proofs of this. One proof, which uses a special rational approximation to each real number, is elementary. The other, which uses the unique factorization property of Gaussian integers, is algebraic. I will also talk about the condition for an integer to be a sum of three squares and give a sketch of its proof. As a direct corollary, we will see every positive integer is a sum of four squares.

We introduce the equation, and then sketch the main ideas needed to show local well-posedness on in the Coulomb gauge for initial data sufficiently small in for $s > \frac{1}{4}.$

We will see that combinatorics is the limit of linear algebra over the field with q elements as q approaches 1. We will then examine one proposed candidate for what this "field with one element" ought to be.

This talk will be an elementary discussion of Clifford Algebras, Spin groups and ultimately Spinors. Next, week I might prove a theorem or two and talk about applications to quantum mechanics, but this week's talk will be about basic definitions and examples, and trying to build some intuition.

Cellular DNA knotting is driven by DNA compaction, topoisomerization, supercoiling-promoted strand collision, and DNA self-interactions resulting from transposition, site-specific recombination, and transcription. Type II topoisomerases are the ubiquitous, essential enzymes that interconvert DNA topoisomers to resolve knots. These enzymes pass one DNA helix through another by creating an enzyme-bridged transient break in the DNA. Explicitly how type II topoisomerases recognize their substrate and decide where to unknot DNA is unknown. The three main investigations to be discussed are the cellular effects of knotting, the biophysics of knotting/unknotting and the unknotting mechanism of human topoisomerase II.

We'll motivate the definition of affine schemes, with lots of examples. If time permits, we will talk about arbitrary (not affine) schemes.

I will give a casual introduction to the 'cousins' of hyperbolic space: the complex, quaternionic, and Cayley hyperbolic spaces. The various finite volume quotients of theses spaces, including hyperbolic space, arise from lattices in the so-called 'real rank one' groups. I will describe how popular questions about hyperbolic 3-manifolds can be generalized to these model spaces and what is known in the different settings.

We study smooth families of symplectic diffeomorphisms that have invariant manifolds arising from a perturbation of a saddle connection. Using a variational approach, it is possible to give a local description of the stable and unstable manifolds for a hyperbolic fixed point. We use this in order to construct a Melnikov potential. We will explore the properties of this function.

Adaptive methods are of cruical importance to efficiency in scientific computing. However, it is still very common that strategies for adaptation used in practice are heuristic or less well supported by analysis. In this talk we review some advances with a focus on time-stepping, where we show that ideas from control theory and digital signal processing can be used to construct efficient techniques for adaptive time-stepping, and that these techniques have an impact on the regularity of the computed solutions as well as on the stability of the computed results. If time permits, we will also have a look at new adaptive methods for Hamiltonian problems, which require that the adaptivity is time-reversible and symmetric, without losing important conservation properties and long-time behavior.

How do you write down a juggling pattern? I'll talk about a system of notation that partially answers this question, and led to the discovery of many previously unknown patterns, as well as some interesting combinatorial problems. There will be many demonstrations. No prior juggling experience required.

Abstract: A rarefied gas flow in a two-dimensional curved channel, driven by a pressure gradient imposed in the gas and/or by a temperature gradient imposed along the channel walls, is investigated on the basis of kinetic theory. Under the assumption that the width of the channel is much smaller than the length scales of variation of the pressure, temperature, and curvature of the channel walls along the channel, a one-dimensional convection-diffusion model is derived by a formal asymptotic analysis based on the BGK model of the Boltzmann equation with the diffuse-reflection boundary condition. The effect of channel curvature and that of gas rarefaction manifest themselves through the coefficients in the convection-diffusion model. A numerical database of these coefficients is constructed by solving basic flow problems along a circular ring. The connection conditions at the junction where the curvature is not continuous are also derived. As an application of the model, a pumping system (i.e., a variant of the Knudsen pump) using a snake-shaped channel with a periodic structure and a periodic temperature distribution along the channel walls is considered, and the compression ratio as well as the pressure distribution along the channel is obtained for the system with many (100) units. The accuracy of the model is confirmed by a full two-dimensional simulation based on the BGK model when the number of the unit is small.

Since gaining the abilities to sequence genomes and generate giant data sets from biological systems, the question has now become how to use and interpret this data. How is this information linked to the characteristics of living organisms? In this talk we'll explore how to begin linking mathematically interesting parts of the transcriptional regulatory network of yeast with their biological significance.

For dimension one and two, a proof of Sharp Strichartz inequality was given recently by Foschi. In this talk, we present, a new representation for the space time norm of solutions of the free Scrodinger equation in dimension one and two in terms of an orthogonal projection operator. As a result of the property that orthogonal projections do not increase the norm, the sharp Strichartz inequality follows." This is a joint work of Hundertmark and Zharnitsky.

Dirichlet's Unit Theorem describes the structure of the unit group in the ring of integers of a number field. Examples of its use will be discussed.

Dominic Joyce posted a paper ( http://arxiv.org/abs/0707.3572 ) which is background for promised proofs of some important conjectures about Gromov-Witten invariants. I will discuss this.

We present a lagrangian proof for the existence of invariant KAM circles. The persistence of such circles shows the persistence of quasi-periodic equilibrium solutions in the extended Frenkel-Kontorova model. The extension we consider states that one particle interacts with infinitely many neighbors with some decay condition. The models that consider interactions with a finite number of neighbors have equilibrium configurations that are equivalent to orbits of singularly perturbed standard maps.

We will discuss systems of Frobenius linear difference equations and their solutions, over function fields of positive characteristic. We will then investigate their applications to the study of Drinfeld modules and characteristic p transcendence problems.

There is a remarkable closed form expression for the Riemann-Zeta function for $2Z_+$ which involves both Bernoulli numbers and . In proving this result we will represent functions as an infinite product of linear polynomials as well as derive a few elegant expressions of functions as infinite sums and products.

We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a quasi-variational inequality and it is across the optimal stopping boundary. Our proof only uses the classical theory of parabolic partial differential equations of Friedman and does not use the \emph{the theory of vicosity solutions}, since our proof relies on constructing a sequence of functions, each of which is a value function of an optimal stopping time for a \emph{diffusion}. (Also, the previous proof holds only for a certain range of parameters.) The sequence is constructed by iterating a functional operator that maps a certain class of convex functions to smooth functions satisfying variational inequalities (or to value functions of optimal stopping problems for a geometric Brownian motion). The approximating sequence converges to the value function exponentially fast, therefore it constitutes a good approximation scheme, since the optimal stopping problems for diffusions can be readily solved using the well-know numerical schemes such as SOR.

Consider the Cauchy problem for a system of viscous conservation laws, with a solution consisting of a viscous shock layer connecting two smooth regions. We expect the time dependent behavior of such a solution to involve two processes. One process consists of the large-scale evolution of the solution. This process is well modeled by the corresponding inviscid system. The second process, which is much faster than the first, is the adjustment in shape and position of the shock layer to the large-scale solution. During this process the end states are essentially constant. In order to analyze stability we separate the two processes. To isolate the fast process we consider the region close to the shock layer. The equations are augmented with special boundary conditions, which reflect the slow change of the end states. For this problem we show that perturbations decay exponentially fast.

There comes a time in the life of a Junior Number Theorist in which he or she might have to face the following fact: they need to know some geometry in order to make good progress. In this talk, I will give a very basic overview (assuming only some knowledge of prelim analysis, topology and algebra) of some geometry that shows up in number theory, and I will focus specifically on the case of elliptic curves. Periods, the Hodge decomposition, Picard-Fuchs equations and monodromy will make brief appearances, and I hope to even prove some theorems if time permits. Also, I will try to explain why I specifically care about these things and what they have to do with my research.

I will start by stating some general facts about spinors and spin groups, but I will spend most of the time working with the two-spinor calculus on Lorentzian manifolds. The double cover of the proper Lorentz group is SL(2,C). As a result, this allows a formalism in which the spinors are just complex 2-vectors. We'll investigate this.

I will discuss some recent results of Lackenby and Long-Lubotzky-Reid that indicate a connection between Property and some virtual conjectures in low-dimensional topology.

We present results on the crossings and nestings for matchings and set partitions, which are generalizations of monotone subsequences in permutations. By a variant of the RSK algorithm from algebraic combinatorics, matchings and set partitions are in one-to-one correspondence with oscillating/vacillating tableaux, which are certain random walks in the Hasse diagram of the lattice of integer partitions. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings of [2n]. Enumeration on non-crossing matchings will be discussed. And we will extend the result to other geometric configurations.

In my talk I will discuss some topics in hyperbolic geometry, including models for the hyperbolic plane, hyperbolic isometries, and various ways to represent them via matrix groups.

We will talk about a "boundary obstacle problem", i.e. the minimization of the Dirichlet integral among all functions that lie above a given obstacle in the boundary of the domain. Such a problem appears in topics such as elasticity, temperature control and modeling of semi-permeable membranes. The optimal regularity of the solution was proven by Caffarelli and Athanasopoulos by exploiting the quasi-convexity of the solutions and a monotonicity formula, this problem is also closely related to the fractional Laplacian. We will discuss the proof of the optimal regularity and current work in this topic.

The aim of this talk is to describe several methods for numerically approximating the integral of a multivariate highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a method developed by Levin as a point of departure we will construct a new method that uses the same information as a Filon-type method, and obtains the same asymptotic order, while not requiring moments. This allows us to integrate over nonsimplicial domains, and with complicated oscillators. We also present a method for approximating oscillatory integrals with stationary points.

Though modular forms are perhaps most important to number theorists, they appear as well in topology, physics, and nature, and their theory uses topology, complex analysis, and both the analytic and algebraic sides of number theory. In this talk we will give an introduction to modular forms. We will begin with their definitions, and cover the central elements of the theory up to and including Hecke operators. There will be key motivating examples throughout. This talk is self contained and is intended to be an introduction. This is the first talk of two and will serve as an introduction for the background that will be assumed in the second.

A pdf of the paper is available here: http://www.ma.utexas.edu/users/jmann/LMZC_JMB06.pdf

In this talk, we are going to discuss the deformation of the surface in the inward normal direction with the speed proportional to Guass or Scalar Curvature. The equation for the graph will be a fully nonlinear and degenerate when one of the principle curvature becomes zero. We will discuss the optimal regularity and the existence of classical solutions.

Dunfield proved that a small hyperbolic knot in a homotopy sphere has a non-integral boundary slope whenever there is a framing $<\alpha, \beta>$ for such that is cyclic. While this result may be categorized as classical 3-manifold topology, the proof relies on ideas from algebraic geometry. In my talk I will discuss character varieties and prove Dunfield's theorem.

I will present a computational method for simulating the dynamics of atomic systems on time scales much longer than can be simulated with classical dynamics. In this method, possible reaction mechanisms available to the system are found with a minimum-mode following algorithm. Rates are estimated using harmonic transition state theory, and the system is propagated in time according to the kinetic Monte Carlo algorithm. The method can be run in parallel, and a distributed computing system has allowed us to simulate the deposition and growth of metal surfaces over experimental time scales using an embedded atom potential. Our algorithm is efficient enough to model the evolution of systems with ab-initio forces as well, for which I will show several examples, including metal cluster formation on oxide surfaces.

A problem in twisted K-theory is to realize twisted K-groups generally by means of finite-dimensional geometric objects, like vector bundles. I will talk about an answer to the problem by using a twisted version of Furuta's generalized vector bundle and a notion of a finite-dimensional approximation of Fredholm operators.

Volume-preserving maps arise from the flow of an incompressible fluid or of a magnetic field. An understanding of their dynamics is important for the study of the mixing of passive tracers, but is also an interesting generalization of the dynamics of area-preserving or symplectic maps. I will show that a quadratic, volume-preserving map arises as a fundamental model analogous to Henon's two-dimensional map in three distinct situations: this model is the "hydrogen atom" of three- dimensional dynamics. Recently we obtained it as the normal form for a bifurcation of a fixed point with a triple-one multiplier. I will discuss dynamical phenomena such as the formation and break-up of invariant circles and tori, the creation of "strings of pearls," and chaotic transport due to heteroclinic intersection of two dimensional invariant manifolds.

We will introduce the Szemeredi Regularity Lemma and go through an example of how it is used to find substructures in large graphs.

(This is joint with Nonlinear Dynamics.)

Viscous flows in thin domains are encountered in a great number of practical situations. Think, for example, of flows in gas or pipelines, in fluid film bearings, in capillaries, or in oceans and in atmosphere. It is natural to expect that in these situations the governing equations of motion are approximated by simpler, lower-dimensional models leading to solutions which are precise only asymptotically as the width of the domain tends to zero. In his seminal paper in 1886, Osborne Reynolds proposed a model for the main pressure distribution in lubrication. This model, the classical Reynolds equation, even though derived under the assumption that the inertial and curvature effects are negligible, makes a reasonable (first-order) approximation for many, but not all, problems in hydrodynamic lubrication. In our work, we derive an equation, referred to as modified nonlinear Reynolds equation, for higher-order approximation of pressure (and consequently of velocity field) in hydrodynamic lubrication. This equation is a nonlinear second-order differential equation for a scalar unknown, which approximates the first two terms in the asymptotic expansion of the pressure field, and, in contrast to the classical Reynolds' equation, takes into account both the inertial and the curvature effects. The equation is justified with optimal integral and pointwise error estimates. Up to now, this kind of optimality has only been achieved for small fluxes and in simpler geometrical situations. Here, the flux, equivalently the Reynolds number, can be as large as possible within the limits of dimension reduction and the optimal estimates result from an accurate analysis of the auxiliary divergence equation.

Bridge number for knots in the 3-sphere is a classical knot invariant introduced by Schubert more than 50 years ago. More recently it has been generalized to knots in any manifold by using a Heegaard surface for the manifold in the place of a bridge sphere. We will discuss some results about bridge surfaces that are analogous to results about Heegaard splittings as well as some of the behaviors of bridge surfaces that seem to have no Heegaard splitting analogs.

We will discuss again the Signorini problem and its optimal regularity in a more rigorous manner than we did last week. The optimal regularity follows from two important lemmas that we will discuss in detail: first that solutions to the thin obstacle problem are semi-convex in the tangential directions to the free boundary and then that the normal derivative of the solution satisfies a monotonicity formula.

is the (1D) continued fraction. The 's can be connected to geometry, via approximating a line L of slope by a sequence of bases of , with 'update matrices' : that is, so that ;. One need only use 2 such 's, to wit and . Then each is just the number of one of these that are needed before the other occurs. CF's are useful for many problems in number theory, in particular, using the ideas of 'best approximants' and 'poorly approximable' numbers lead
1) to special types of (real) numbers useful in mathematics.
2) As another example, one can decide whether 2 'Pisot' 2 by 2 matrices are conjugate over the integers; and
3) Can show that given A as in (2) there is a B in GL(z,2) such that ; has all entries of the same sign. Though mostly other authors have focused on problems of type (1) in their generalizations to ``higher dimensional continued fractions'', we propose here a theory of HDCF useful in problems of the second two types. In particular, this solves these problems in dimension 3. The fundamental construction is a cell complex homotopy equivalent to the 2 skeleton of the 3 dimensional torus, tying arithmetic to geometry.

At the center of modular forms is the theory of Hecke operators. Hecke operators are a commutative algebra of operators acting on the space of modular forms of weight . This space has a canonical basis of simultaneous eigenvectors of all the and these particular modular forms have very special Fourier coefficients, L functions, and connections to elliptic curves. We will begin with the definition of the Hecke operators and explain some key results. If time permits, we will explain how this connects to the Taniyama-Shimura-Weil Theorem proved by Wiles for Fermat’s Last Theorem, and show how to get an elliptic curve from a cusp form. There will be a recap of what we discussed last week as well. The notes from last week's talk can be found at the following link: http://www.math.utexas.edu/users/khopkins/JNTTalkModFormsFirstHalf.pdf

We will discuss the regularity theory of the Monge-Ampere equation det D^2u=f(x) where f(x) is in VMO^\psi spaces defined in terms of sections.

After reviewing last week's talk, I will finish what I started.

We consider a few modeling and computation problems that is related to incoherent imaging in highly heterogeneous (random) media. We show that such imaging problems can be treated as inverse problems related to the linear transport equations. Numerical solutions of those inverse problems will then be discussed. Reconstructions from both synthetic and experimental data will be presented.

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, and . In our normalization, the indices are allowed to be half- integers as well as integers, and the ``polynomials'' actually live in . We will show that the rational functions and are very remarkable from the point of view of Grothendieck's dessins d'enfants. The fibers of these rational functions are likewise very remarkable from the point of view of algebraic number theory. For example, for the fiber of the second function above is given by a degree 15875 polynomial in with discriminant and Galois group the entire symmetric group .

Articles: Aspinwall, "D-branes on Calabi-Yau manifolds", Progress in String Theory, TASI 2003 Lecture. Notes (2005), World Scientific. Bridgeland, "Stability conditions on triangulated categories", Ann. of Math. 2007, Vol 166, No.2. Kontsevich, "Donaldson-Thomas invariants", 2007 MPIM Mathematische Arbeitstagung, http://www.mpim-bonn.mpg.de/preprints/send?bid=3352.

In this talk will define strongly -regular rings and show how that every strongly -regular element can be written as the sum of an idempotent, a unit and a nilpotent element, all of which commute with one another. That every strongly -regular element is strongly clean is a consequence of this result.

Building convergent numerical schemes for fully nonlinear partial differential equations is a challenge. In the case of first order equations, schemes have been built based on ideas from Conservation Laws. Recently several authors have tried adapting techniques from fluid mechanics or finite elements to solve second order equations. These attempts have had limited success. We will present recent work on bulding convergent difference schemes for degenerate and fully nonlinear elliptic Partial Differential equations. These equations include: the obstacle problem, the infinity Laplacian, the equation for motion of level sets by mean curvature, the Monge-Ampere equation, the Pucci maximal and minimial equations, an equation for the convex envelope. Earlier work emphasized building convergent schemes over fast or accurate solvers. Since the resulting algebraic equations are nonlinear and non-differentiable, attempts to build implicit solvers have met with little success. As a result, the solution methods have been iterative, and consequently suffer from CFL type time step limitations. Recent work has explored technique to improve accuracy as solution speed. Remarkably, many of the fully nonlinear schemes can be reformulated as Linear Programs (LPs). We will discuss hybrid schemes for better accuracy, and implicit solvers for fully nonlinear equations.

TBA

We will provide a construction of traveling waves in a lattice model through a reduction to a center manifold technique. These models coming from solid state physics are known to play an important role in mechanisms involved in the propagation of energy. This is joint work with Yannick Sire.

I will give an overview of Guass' class number conjecture and present Estermann's proof of Siegel's Theorem. Special attention will be paid to the entrance of the ineffective constant in the proof.

(Part 1 of 2.) These talks will feature a colorful and interrelated miscellany of items that have made repeated appearances in the seminar this semester, including K-theory, Clifford algebras, spinors, and characteristic classes.

I'll explain some definitions of a geometrically finite Fuchsian group, and why they're equivalent. I'll extend them to higher dimensions, and give examples showing why some formulations can be more useful than others there.

In this talk, I will discuss three topics about the DG method. I will describe two recently developed discontinuous Galerkin schemes for solving the Hamilton-Jacobi equation and equations with higher order derivatives, respectively. Both schemes do not need to rewrite the original equation into several other equations using auxiliary variables to compute the solution. For the Hamilton-Jacobi equation, our scheme works well when the Hamiltonian is convex or linear. An entropy correction is needed when the entropy condition is violated. For equations with higher order spatial derivatives, our scheme can give provable L2 stability and accuracy for a very general class of equations. Numerical tests show that this scheme is (k+1)-th order accurate when using piecewise k-th polynomials, under the condition that k+1 is greater than or equal to the order of the equation. I will also present some recent work on the superconvergence and time evolution of DG method for conservation laws when essentially upwind fluxes are used. This is a joint work with Professor Chi-Wang Shu.

Singular vectors in R^d correspond to divergent trajectories of the homogeneous flow on SL(d+1,R)/SL(d+1,Z) induced by the one parameter subgroup diag(e^t,...,e^t,e^&ob;-dt&cb;) acting by left multiplication. In this talk, I will sketch a proof of the following result: the Hausdorff dimension of the set of singular vectors in R^2 is 4/3. (Alternatively, the set of points lying on divergent trajectories of the homogeneous flow on SL(3,R)/SL(3,Z) has Hausdorff dimension 7 and 1/3.) The main idea involves a multi-dimensional generalisation of continued fraction theory from the perspective of the best approximation properties of convergents. As an application, we answer a question of A.N. Starkov regarding the existence of slowly divergent trajectories.

For non-negative integers , let be the unique modular function that is holomorphic on the upper half-plane and with Fourier expansion . Asai, Kaneko, and Ninomiya showed that , where as usual is the weight Eisenstein series and is the unique normalized weight cusp form. Ahlgren obtained similar formulas on higher level, genus , congruence subgroups, with replaced by the corresponding Hauptmodul. We generalize these results to congruence subgroups for which the underlying modular curve is hyperelliptic. In all cases, we also show how to obtain similar results starting from any meromorphic modular form of weight . (From this point of view, the above formula corresponds to ).

At the beginning of 80's Simon Donaldson introduced new invariants of 4-dimensional Rimannian manifolds by counting critical points of the Yang-Mills functional on the space of connections. At the end of 90's his student Richard Thomas constructed new invariants of 3-dimensional complex Calabi-Yau manifolds by counting critical points of holomorphic Chern-Simons functional. The latter invariants and their versions for other gauge theories are called Donaldson-Thomas invariants. > From the algebraic point of view Donaldson-Thomas invariants should ``count" stable objects in triangulated categories. In my talk I plan to discuss our recent work with Maxim Kontsevich, where we made this idea more precise. We introduced "motivic" version of Donaldson-Thomas invariants by encoding them as invertible elements of certain non-commutative algebras similar to quantum tori of Connes. As a byproduct we obtained a new type of ``wall-crossing" formula, which describes the behavior of the invariants when the stability structure on the category gets changed. If time permits I will discuss the relation to cluster transformations as well as to the count of BPS states in N=2 4d super Yang-Mills theory.

Steganography is the method of hiding information beyond the standard level of encryption so that the presence of the hidden data is not detectable. I will present an overview of a few topics in Harmonic Analysis and will discuss how Fourier transforms and wavelets pertain to steganography of digital information.

In the theory of differential games with asymmetric information the value function u(x,t,p,q) (where p and q are parameters) has to "satisfy" a parabolic equation in (x,t) for any value of the parameters and at the same time remain convex in and concave in . We shall explain the proper framework of analysis for this equation, give some representation formula of the solution, and discuss open problems.

We discuss several one-dimensional symmetry results for global solutions of (possibly singular or degenerate in some cases) elliptic equations, inspired by a conjecture of De Giorgi. In particular, symmetry results are given under the assumption that either the profiles at infinity are two-dimensional, or that one level set is a complete graph, or that the solution is quasi-minimal. In dimension two and three, symmetry results will be obtained under very mild assumptions.

In this talk, I will introduce J.M. Bony's notion of a paraproduct between two functions, and present several linearization results of Y. Meyer which utilize this construction.

We will briefly introduce the concept of arcs and complete arcs. We will then talk about the Giulietti-Ughi and state some results we obtained. If time permits we will state and prove a theorem.

This lecture blends ideas from probability theory, PDE’s, numerical analysis and physical reasoning in the style of modern applied mathematics. First elementary concepts of information theory involving the lack of information in one probability measure relative to another are introduced. These ideas are then applied to give a unified explanation of the competing equilibrium statistical theories for coherent structures in fluids. The potential application of these ideas to predict coherent structures in flows with random forcing and dissipation is then discussed briefly including the first theorem (Xiaoming Wang and Majda, CPAM 2006) justifying the approach. All of this is utilized to develop a successful prediction for the exact location and structure of Jupiter’s Red Spot in agreement with the Voyager Mission of the 1970’s by utilizing a physically based prior encoding the key features of small scale observations from the Galileo mission of the 1990’s. An apriori independent justification of this overall strategy is presented through novel numerical methods for detecting statistically relevant conserved quantities in fluid flow (Abramov and Majda, 2003, 2004.) The lecturer’s recent book with X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flow, Cambridge Univ. Press (2006) discusses in detail most of the material presented here.

Geophysical flows are a rich source of novel problems for applied mathematics and the contemporary theory of partial differential equations. The reason for this is that many physically important geophysical flows involve complex nonlinear interaction over multi-scales in both time and space so developing simplified reduced models which are simpler yet capture key physical phenomena is of central importance. In mid-latitudes, the fact that the rotational Coriolis terms are bounded away from zero leads to a strict temporal frequency scale separation between slow potential vorticity dynamics and fast gravity waves; this physical fact leads to new theorems justifying the quasi-geostrophic midlatitude dynamics even with general unbalanced initial data for both rapidly rotating shallow water equations and completely stratified flows. At the equator, the tangential projection of the Coriolis force from rotation vanishes identically so that there is no longer a time scale separation between potential vortical flows and gravity waves. This has profound consequences physically that allow the tropics to behave as a waveguide with extremely warm surface temperatures. The resulting behavior profoundly influences longer term mid-latitude weather prediction and climate change through hurricanes, monsoons, El Nino, and global teleconnections with the mid-latitude atmosphere. How this happens through detailed physical mechanisms is one of the most important contemporary problems in the atmosphere-ocean science community with a central role played by nonlinear interactive heating involving the interaction of clouds, moisture, and convection. The variable coefficient degeneracy of the Coriolis term at the equator alluded to earlier leads to both important new physical effects as well as fascinating new mathematical phenomena and PDE’s. In this equatorial context, new multi-scale reduced dynamical PDE models are even relatively recent in origin. After a brief discussion of the observational record as background, this lecturer surveys the remarkable new hyperbolic systems that have emerged recently in applications including their physical properties, applied mathematical and rigorous mathematical theory. These last topics include novel relaxation limits for climate models with active moisture and new singular limits for hyperbolic PDE’s with variable coefficients. All of the references in this lecture can be found at http://www.math.nyu.edu/faculty/majda/.

We'll talk about what spectral sequences are, how they arise, and why we might care about them. Then we'll discuss a particular example from Khovanov homology, used by Rasmussen to give a new proof of a conjecture of Milnor.

Whether smooth solutions of the 3D incompressible Euler equations spontaneously develop finite time singularities is a long-standing open problem of great interest to both mathematical and fluid mechanical communities. In this talk, I will discuss recent results on this problem obtained with T. Y. Hou, J. Deng, and R. Li. We investigate possible cancellations in the nonlinear term that prevent the formation of finite time singularities. I will also show numerical results supporting the existence of such cancellations.

We discuss ergodic theorems for semisimple groups and lattices and apply them to derive some effective counting and equidistribution results. Based on joint work with Amos Nevo.

This year's crossword puzzle: "GETTING TO KNOW YOU - WHAT'S IN A NAME?" Prizes to the person or team that solves the puzzle first. A special thank you to Dr. Karen Uhlenbeck for the prizes. MOVIE: A HILL ON THE DARK SIDE OF THE MOON THE STORY OF SONYA KOVALEVSKAYA Host: Dr. Fernando Rodriguez-Villegas Special Introduction by: Dr. Natasa Pavlovic RSVP TO NANCY LAMM.

It would be nice to think that arithmetic algebraic geometry had something useful to say about the world we live in. The talk will concern the combinatorics of graphs and the physics of Feynman amplitudes

This lecture discusses systematic mathematical strategies for low-dimensional stochastic mode reduction for turbulent large dimensional dynamical systems and their application to modelling low frequency weather dynamics and climate change. A remarkable fact of Northern Hemisphere low frequency variability is that it can be efficiently described by only a few teleconnection patterns that explain most of the total variance. These few teleconnection patterns not only exert a strong influence on regional climate and weather, they are also related to climate change. These properties of teleconnection patterns make them an attractive choice as basis functions for climate models with a highly reduced number of degrees of freedom. The development of such reduced climate models involves the solution of two major issues: 1) how to properly account for the unresolved modes, also known as the closure problem; and 2) how to define a small set of basis functions that optimally represent the dynamics of the major teleconnection patterns. In this lecture examples of stochastic mode reduction are discussed ranging from an explicit solvable pedagogical example with three modes to a prototype atmospheric general circulation model with a thousand degrees of freedom where an effective reduced stochastic model with only ten low frequency modes captures the statistical dynamical behavior. A controversial topic in the recent climate modeling literature is the fashion in which metastable low-frequency regimes in the atmosphere occur despite nearly Gaussian statistics for these planetary waves. Here a simple 57-mode paradigm model for such metastable atmospheric regime behavior is introduced and analyzed through hidden Markov model (HMM) analysis of the time series of suitable low-frequency planetary waves. The analysis of this paradigm model elucidates how statistically significant metastable regime transitions between blocked and zonal statistical states occur despite nearly Gaussian behavior in the associated probability distribution function and without a significant role for the low-order truncated nonlinear dynamics alone; turbulent backscatter onto the three-dimensional subspace of low-frequency modes is responsible for these effects. It also is demonstrated that suitable stochastic mode reduction strategies, which include both augmented cubic nonlinearity and multiplicative noise, are also capable of capturing the metastable low-frequency regime behavior through a single stochastic differential equation compared with the full turbulent chaotic 57-mode model. This feature is attractive for issues such as long-term weather predictability. Research papers regarding most of the research here can be found on Majda’s faculty website: http://www.math.nyu.edu/faculty/majda

We will prove the Gauss-Bonnet Theorem for orientable surfaces, which is a famous and remarkable local-global result in differential geometry that relates Gaussian curvature of a surface to its Euler characteristic.

Ordinary continued fractions CF have been generalized to higher dimensions, HDCF's since Jacobi and Perron's thesis, and in many different ways. No one way is as useful as CF's. We start with showing how 2 of these ways are generalizations of CF's and talk about a way that is useful for solving the following problems: call a matrix 'Pisot' if it has a real eigenvalue and all others inside the unit circle. 1. If is a Pisot matrix there exists such that has all entries 2. Given 2 Pisot matrices, there is a finite procedure for deciding whether they are conjugate over the integers.

I will first introduce the notion of stability for steady solutions to Euler's equation, then discuss how looking at fluid motion from a geometric point of view inspired some questions that (I think) could only be answered analytically. This will motivate separately looking for instabilities in each of the two pieces of an orthogonal decomposition of the solution space of Euler's equation.

In recent years there have been a number of substantial advances in numerical techniques for solving the wave equation and Maxwell's equations in the time domain. These include: (i) Efficient methods for truncating unbounded domains which can provide arbitrary accuracy at low cost; (ii) Fast methods for evaluating the standard integral solution formulas, akin to the fast multipole method in the frequency domain; (iii) Discretization strategies focusing on techniques for using high order/high resolution approximations in complex geometry. I will review these new developments and discuss ideas for their improvement and extension to other wave propagation problems.

We will begin by introducing the Weil height of algebraic numbers and the Mahler measure of polynomials in one variable. We will then describe Lehmer's problem and the most important existing results relating to it. We will then review normalizers of stabilizers desribe their relationships to the measure and height. Time permitting, we will describe my research results.

In the 1960's "classical", or "zero-th order" knot signatures were defined and shown to have special behavior on slice knots. In the 1970's Casson and Gordon defined what we shall call "first-order" signatures that were also shown to obstruct a knots being a slice knot. We define a partial derivative (and antiderivative) of a knot. In terms of this notion we define "second-order" signatures of a knot and show that they obstruct a knots being a slice knot. In some situations we can take more derivatives and define "n-th order" signature obstructions. These are applied to show that certain increasingly subtle families of knots contain no slice knots.

I will present Korner's recent construction of Besicovtich sets via Baire's Theorem and give an overview of applications and related open questions in Harmonic analysis. This talk will assume little beyond introductory real analysis.

Cochran, Orr and Teichner defined a filtration of the smooth and topological knot concordance groups. The quotients of successive terms of these filtrations are abelian groups that reflect the existence of increasingly subtle non-slice knots. We prove that each such quotient has infinite rank.

Abstract: Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss results on Donaldson's program of extending the Calabi-Yau theory to symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.

Given a separable polynomial q with coefficients in a field K, we study the trace form on the vector space K[x]/q. When it is real and definite, its generating function w is a rational function that gives rise to some nice results on the interlacing of the real roots of q and qw. This opens the door to the mystery of the p-adic world: how would this phenomena generalize over there?

This will be more conjecture than theorem meant to bring together different ideas, including applications to field theory and string theory. I will explain how the "derived philosophy" in this context yields stronger information than the usual theorems.

We will analyze the evolution of weakly convex surfaces in with flat sides by the harmonic mean curvature flow. We establish short time existence as well as the optimal regularity of the surface and show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class , for some integer and , remains in the same class.

Today will be our end of the semester problem session. This is an opportunity for anyone to get up and speak briefly about a problem, big or small, the junior topology audience may appreciate.

We will first prove some general existence theorems about "heavy points" in a measure preserving system: points whose ergodic averages consistently dominate their expected value. We will then proceed to some examples of this behavior, contrasting the existence of such behavior in dynamical systems with the absence of such objects in other settings, and contrasting the ease of an "existence proof" with the difficulty of a constructive proof.